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Theorem iunxun 3726
Description: Separate a union in the index of an indexed union. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
Assertion
Ref Expression
iunxun x (AB)𝐶 = ( x A 𝐶 x B 𝐶)

Proof of Theorem iunxun
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 rexun 3117 . . . 4 (x (AB)y 𝐶 ↔ (x A y 𝐶 x B y 𝐶))
2 eliun 3652 . . . . 5 (y x A 𝐶x A y 𝐶)
3 eliun 3652 . . . . 5 (y x B 𝐶x B y 𝐶)
42, 3orbi12i 680 . . . 4 ((y x A 𝐶 y x B 𝐶) ↔ (x A y 𝐶 x B y 𝐶))
51, 4bitr4i 176 . . 3 (x (AB)y 𝐶 ↔ (y x A 𝐶 y x B 𝐶))
6 eliun 3652 . . 3 (y x (AB)𝐶x (AB)y 𝐶)
7 elun 3078 . . 3 (y ( x A 𝐶 x B 𝐶) ↔ (y x A 𝐶 y x B 𝐶))
85, 6, 73bitr4i 201 . 2 (y x (AB)𝐶y ( x A 𝐶 x B 𝐶))
98eqriv 2034 1 x (AB)𝐶 = ( x A 𝐶 x B 𝐶)
Colors of variables: wff set class
Syntax hints:   wo 628   = wceq 1242   wcel 1390  wrex 2301  cun 2909   ciun 3648
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-iun 3650
This theorem is referenced by:  iunsuc  4123  rdgisuc1  5911  oasuc  5983  omsuc  5990
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